Simplify (7+8i)^2

${(7 + 8 i)}^{2}$

Step 1

Rewrite ${(7 + 8 i)}^{2}$ as $(7 + 8 i) (7 + 8 i)$ .

$(7 + 8 i) (7 + 8 i)$

Step 2

Expand $(7 + 8 i) (7 + 8 i)$ using the FOIL Method.

Apply the distributive property.

$7 (7 + 8 i) + 8 i (7 + 8 i)$

Apply the distributive property.

$7 \cdot 7 + 7 (8 i) + 8 i (7 + 8 i)$

Apply the distributive property.

$7 \cdot 7 + 7 (8 i) + 8 i \cdot 7 + 8 i (8 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$49 + 7 (8 i) + 8 i \cdot 7 + 8 i (8 i)$

Multiply $8$ by $7$ .

$49 + 56 i + 8 i \cdot 7 + 8 i (8 i)$

Multiply $7$ by $8$ .

$49 + 56 i + 56 i + 8 i (8 i)$

Multiply $8 i (8 i)$ .

Multiply $8$ by $8$ .

$49 + 56 i + 56 i + 64 i i$

Raise $i$ to the power of $1$ .

$49 + 56 i + 56 i + 64 (i^{1} i)$

Raise $i$ to the power of $1$ .

$49 + 56 i + 56 i + 64 (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$49 + 56 i + 56 i + 64 i^{1 + 1}$

Add $1$ and $1$ .

$49 + 56 i + 56 i + 64 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$49 + 56 i + 56 i + 64 \cdot - 1$

Multiply $64$ by $- 1$ .

$49 + 56 i + 56 i - 64$

Subtract $64$ from $49$ .

$- 15 + 56 i + 56 i$

Add $56 i$ and $56 i$ .

$- 15 + 112 i$